Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices.
Contents
Definition and basic properties
An n-dimensional multi-index is an n-tuple
of non-negative integers (i.e. an element of the n-dimensional set of natural numbers, denoted
For multi-indices
where
where
Some applications
The multi-index notation allows the extension of many formulae from elementary calculus to the corresponding multi-variable case. Below are some examples. In all the following,
Note that, since x+y is a vector and α is a multi-index, the expression on the left is short for (x1+y1)α1...(xn+yn)αn.
For smooth functions f and g
For an analytic function f in n variables one has
In fact, for a smooth enough function, we have the similar Taylor expansion
where the last term (the remainder) depends on the exact version of Taylor's formula. For instance, for the Cauchy formula (with integral remainder), one gets
A formal linear N-th order partial differential operator in n variables is written as
For smooth functions with compact support in a bounded domain
This formula is used for the definition of distributions and weak derivatives.
An example theorem
If
Proof
The proof follows from the power rule for the ordinary derivative; if α and β are in {0, 1, 2, . . .}, then
Suppose
For each i in {1, . . ., n}, the function
for each